Michael Zheng shifted uncomfortably on the base line of Court 13 at the U.S. Open, an outdoor court he had played on as a boy in an elite training program. But this time there was much more at stake, with hundreds of fans crammed in on all four sides, chanting, “Let’s go, Michael,” for the local favorite.

Mr. Zheng, who grew up in Montville, N.J., and is a star player at Columbia University, was playing for the chance to qualify for one of the last spots at the United States Open.

Just like 63 other players in the men’s singles qualifying draw, Mr. Zheng, 20, needed to win three straight matches in the preliminary week of the Open to join the big boys. The 16 players who qualify gain entry into the main draw, where the likes of Carlos Alcaraz will compete for the title and a $3.6 million prize.

But standing across the net from Mr. Zheng on Tuesday was a serious opponent: Kamil Majchrzak, a 28-year-old player from Poland who was once ranked No. 75 in the world and had reached the third round of the U.S. Open main draw in 2019. Mr. Zheng had dropped the first set and was clearly suffering from cramps.

It was already a tricky draw, but Mr. Zheng, who comes from a family of achievers, has excelled at whatever level of tennis he plays. As an 18-year-old junior, he went all the way to the final of the boys’ singles at Wimbledon in 2022. This year, he became the first Columbia player to reach the final of the N.C.A.A. singles tournament, and this month he won his first professional title in a challenger tournament (the minor leagues of tennis) in Mississippi.